# unique subgroups

• Oct 18th 2008, 06:57 PM
dori1123
unique subgroups
Prove that \$\displaystyle A_4\$ has a unique subgroup \$\displaystyle V\$ of order 4.

I know all 12 elements of \$\displaystyle A_4\$, and I know \$\displaystyle V\$ contains \$\displaystyle { 1, (12)(34), (13)(24), (14)(23)}\$, but how do I prove that \$\displaystyle V\$ is a unique subgroup?
• Oct 18th 2008, 06:59 PM
ThePerfectHacker
Quote:

Originally Posted by dori1123
Prove that \$\displaystyle A_4\$ has a unique subgroup \$\displaystyle V\$ of order 4.

I know all 12 elements of \$\displaystyle A_4\$, and I know \$\displaystyle V\$ contains \$\displaystyle { 1, (12)(34), (13)(24), (14)(23)}\$, but how do I prove that \$\displaystyle V\$ is a unique subgroup?

Since subgroups of order \$\displaystyle 4\$ are Sylow \$\displaystyle 2\$-subgroups it follows all subgroups of order \$\displaystyle 4\$ must be conjugate by Sylow's second theorem. Now \$\displaystyle V\$ is a normal subgroup of \$\displaystyle A_4\$. Therefore, it must be the only one.